Since \( is a RKHS, it has the reproducing property, as stated in Definition~\ref {reproducing_property}, so we can write \) f(xj)\( as\begin{equation}f(x_j) = \left\langle f(\cdot ),k(x_j, \cdot ) \right\rangle  = \sum ^n_{i=1} \alpha _ik(x_i, x_j) + \left\langle f_{\perp }(\cdot ),k(x_j, \cdot ) \right\rangle _ \sum ^n_{i=1}\alpha _ik(x_i, x_j)\end{equation}for all \) j{1, ..., n}\(.\\For all \) f\(,\begin{equation}\Omega (\Vert f\Vert _ = \bar{\Omega } \left( \left\Vert \sum ^n_{i=1} \alpha _ik(x_i, \cdot )\right\Vert ^2_ \Vert f_\perp \Vert ^2_) \ge \bar{\Omega }\left( \left\Vert  \sum ^n_{i=1} \alpha _ik(x_i, \cdot ) \right\Vert ^2_)\right.Thus, for any \right.\alpha _i\in \mathbb {R}, the risk functional considered in this theorem, is minimized for f_\perp =0. The same holds for the solution, so the theorem is proved. \\\Box \end{equation}\) The original form of this theorem, provided in [1]}, consisted in the particular case in which \(\ell \)  is the point-wise mean squared loss, and \(\Omega (f) = \Vert f\Vert _2\) . While in the form presented in Theorem REF , provided by [2]}, they drop the loss function restriction, enlarging the applicability to any strictly monotonic increasing loss function.
